On the cohomological dimension of finitely generated modules
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Let $(R,\m)$ be a commutative Noetherian local ring and $I$ be an ideal of $R$. In this paper it is shown that if ${\rm cd}(I,R)=t>0$ and the $R$-module $\Hom_R(R/I, H^t_I(R))$ is finitely generated, then \begin{align*} t={\rm sup}\,\{\dim \widehat{\widehat{R}_{\mathfrak P}}/\mathfrak Q:~&\mathfrak P\in V(I\widehat{R}),\,\,\mathfrak Q\in \mAss_{\widehat{\widehat{R}_{\mathfrak P}}}\widehat{\widehat{R}_{\mathfrak P}}\,\,{\rm and}\\ &\mathfrak P\widehat{\widehat{R}_{\mathfrak P}}=\Rad(I\widehat{\widehat{R}_{\mathfrak P}}+\mathfrak Q)\}. \end{align*} Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R\subset R[X]$ will be included.
Abstract
Let $(R,\m)$ be a commutative Noetherian local ring and $I$ be an ideal of $R$. In this paper it is shown that if ${\rm cd}(I,R)=t>0$ and the $R$-module $\Hom_R(R/I, H^t_I(R))$ is finitely generated, then \begin{align*} t={\rm sup}\,\{\dim \widehat{\widehat{R}_{\mathfrak P}}/\mathfrak Q:~&\mathfrak P\in V(I\widehat{R}),\,\,\mathfrak Q\in \mAss_{\widehat{\widehat{R}_{\mathfrak P}}}\widehat{\widehat{R}_{\mathfrak P}}\,\,{\rm and}\\ &\mathfrak P\widehat{\widehat{R}_{\mathfrak P}}=\Rad(I\widehat{\widehat{R}_{\mathfrak P}}+\mathfrak Q)\}. \end{align*} Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R\subset R[X]$ will be included.
참고문헌 (18)
- K. Bahmanpour, "A note on Lynch's conjecture", Comm. Algebra, 45(6), pp.2738~2745 (2017)[DOI]
- K. Bahmanpour, "On the cofiniteness of local cohomology modules", Proc. Amer. Math. Soc., 136(7), pp.2359~2363 (2008)[DOI]
- K. Bahmanpour, "Cofiniteness of local cohomology modules for ideals of small dimension", J. Algebra, 321(7), pp.1997~2011 (2009)[DOI]
- K. Bahmanpour, "Minimaxness and cofiniteness properties of local cohomology modules", Comm. Algebra, 41(8), pp.2799~2814 (2013)[DOI]
- M. P. Brodmann, "Local Cohomology: an algebraic introduction with geometric applications" (1998), Cambridge University Press
- K. Divaani-Aazar, "Cohomological dimension of certain algebraic varieties", Proc. Amer. Math. Soc., 130(12), pp.3537~3544 (2002)[DOI]
- G. Faltings, "Uber lokale Kohomologiegruppen hoher Ordnung", J. Reine Angew. Math., 313, pp.43~51 (1980)
- G. Ghasemi, "Upper bounds for the cohomological dimensions of finitely generated modules over a commutative Noetherian ring", Colloq. Math., 137(2), pp.263~270 (2014)[DOI]
- A. Grothendieck, "Local Cohomology", Lecture Notes in Math, 862 (1966)
- R. Hartshorne, "Cohomological dimension of algebraic varieties", Ann. of Math., 88, pp.403~450 (1968)[DOI]
- R. Hartshorne, "Affine duality and cofiniteness", Invent. Math., 9, pp.145~164 (1970)[DOI]
- M. Hellus, "Matlis duals of top local cohomology modules and the arithmetic rank of an ideal", Comm. Algebra, 35(4), pp.1421~1432 (2007)[DOI]
- M. Hellus, "Matlis duals of top local cohomology modules", Proc. Amer. Math. Soc., 136(2), pp.489~498 (2008)
- C. Huneke, "On the vanishing of local cohomology modules", Invent. Math., 102(1), pp.73~93 (1990)[DOI]
- H. Matsumura, "Commutative Ring Theory" (1986), Cambridge Univ. Press
- A. A. Mehrvarz, "Arithmetic rank, cohomologal dimension and filter regular sequences", J. Algebra Appl., 8(6), pp.855~862 (2009)[DOI]
- P. Schenzel, "Proregular sequences, local cohomology, and completion", Math. Scand., 92(2), pp.161~180 (2003)[DOI]
- H. Zoschinger, "Minimax modules", J. Algebra, 102(1), pp.1~32 (1986)[DOI]